Additively stable sets, critical sets for the 3k-4 theorem in $\mathbb{Z}$ and $\mathbb{R}$

Abstract

We describe in this paper additively left stable sets, i.e. sets satisfying $\left((A+A)-\inf(A)\right)\cap[\inf(A),\sup(A)]=A$ (meaning that $A-\inf(A)$ is stable by addition with itself on its convex hull), when $A$ is a finite subset of integers and when $A$ is a bounded subset of real numbers. More precisely we give a sharp upper bound for the density of $A$ in $[\inf(A),x]$ for $x\le\sup(A)$, and construct sets reaching this density for any given $x$ in this range. This gives some information on sets involved in the structural description of some critical sets in Freiman's $3k-4$ theorem in both cases.